DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A graded derivation of the algebra of differential forms

Definition

Let $M$ be a smooth manifold and $r \in Z$ , and put $Ω^{j} (M) = {0}$ for $j < 0$ . A degree- $r$ graded derivation of $Ω^{*} (M)$ is an $R$ -linear map $D : Ω^{*} (M) \to Ω^{*} (M)$ such that $D Ω^{k} (M) \subseteq Ω^{k + r} (M)$ for every $k \geq 0$ and $D (α \land β) = D α \land β + (- 1)^{r \deg α} α \land D β$ for all homogeneous forms $α$ and $β$ .

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.